Optics 062 · Diffraction, Fourier optics, and computational imaging

Optical Fourier Processor

A coherent 4f processing bench with eight passive amplitude or phase filters, movable frequency masks, real inverted receiving data, phase contrast and matched correlation.

Object-to-receiver distance / programmed filter radius

4f,a_H

$0.50$

Complete-field throughput / visible crop power

T,P_{\rm crop}

$50\%$

Object sampling / frequency spacing

\Delta x,\Delta\nu

$0.00\pi$

Model regime $\text{valid model regime}$

Validated scientific apparatus

Optical Fourier Processor

Filter index (named buttons below)

h

1

Object index (named buttons below)

o

0

Radius / slit half-width (spatial frequency)

r_\nu

2.50\,\mathrm{mm}^{-1}

Horizontal filter centre

\nu_{cx}

0\,\mathrm{mm}^{-1}

Vertical filter centre

\nu_{cy}

0\,\mathrm{mm}^{-1}

Directional slit long-axis angle

\alpha

0^{\circ}

Programmed disk phase delay

\phi_H

0.50\pi

Coherent monochromatic wavelength

\lambda

532\,\mathrm{nm}

Both lens focal lengths (moves the bench)

f

100\,\mathrm{mm}

Image display gain exponent (not raw data)

e_G

0.50

A tiny frequency mask can erase a stripe or expose an invisible phase object. Start with low versus high pass, then select the stripe preset and move the twin notch onto its two spectrum peaks. A phase disk changes local contrast without changing total transmitted power; matched correlation locates the repeated open-ring glyph. The actual positive-lens 4f bench produces an inverted image. Input, frequency mask, spectra and receiving image share one complex-amplitude computation, not independently painted pictures. Enlarged data stay outside the apparatus. Click the computed spectrum to translate the physical filter. Changing wavelength or focal length reprograms its physical size to hold the selected spatial-frequency coordinates; this is not a fixed manufactured mask. Raw throughput stays separate from logarithmic brightness.

Ideal equal-focal-length unlimited-pupil lenses, synthetic finite coherent field, passive programmed Fourier-plane stop. Complex 512-square zero-padded FFT and point samples; periodic numerical output, not camera pixel integration or intensity-only phase recovery. Fixed logarithmic display and unequal scene scales. No aberrations, incoherence, noise or expert certification.

Start with the essentials

Focus question: How does moving one filter in a Fourier plane rewrite edges, blur, and repeated texture in an image?
One-sentence intuition: Two positive lenses make two forward transforms. Thus $U_{\rm out}(x,y)=\mathcal F^{-1}\{H\widetilde U_{\rm in}\}(-x,-y)$ . The minus coordinates matter: the receiving-plane image is inverted in both transverse directions.

Core mathematical model

Front and back focal planes

\begin{aligned}\widetilde U(\nu_x,\nu_y)&=\iint U(x,y)e^{-2\pi i(\nu_x x+\nu_y y)}\,\mathrm dx\,\mathrm dy\\U_F(X,Y)&=\frac{e^{2ikf}}{i\lambda f}\widetilde U\left(\frac X{\lambda f},\frac Y{\lambda f}\right)\end{aligned}

Object is one focal length before the first ideal lens. The filter is one focal length after it and one before the second lens. The receiver is one behind the second. Common carrier phase does not change intensity.

Passive masks and phase

\begin{aligned}H_{\rm low}&=P\,\mathbf1_{|\boldsymbol\nu-\boldsymbol\nu_c|\le r_\nu}\\H_{\rm high}&=P-H_{\rm low}\\H_{\rm phase}&=P\exp\left(i\phi_H\mathbf1_{|\boldsymbol\nu-\boldsymbol\nu_c|\le r_\nu}\right)\\P&=\mathbf1_{|\boldsymbol\nu|\le14\,\mathrm{mm}^{-1}}\end{aligned}

Band passes radii between half radius and radius. Directional mode is a slit with that half-width. Twin notch blocks both signed centres. Phase disk is programmed at the chosen wavelength, not an unspecified glass depth. Carrier is a Fourier-plane stop; ideal lens pupils remain unlimited.

Power is separate from display brightness

\begin{aligned}T&=\frac{\sum|\widetilde U H|^2}{\sum|\widetilde U|^2}\\I_{\rm out}&=|U_{\rm out}|^2\\B_I&=\min\left(1,\frac{\ln(1+10^{e_G}I/I_{\rm inc})}{\ln5}\right)\end{aligned}

Parseval gives the same transmission from the complete sampled receiving field. The five-millimetre visible crop is not total power. Input, filtered and comparison images share one fixed display mapping; Fourier diagnostics use a separate fixed logarithmic mapping. All raw probes remain linear.

Phase becomes contrast, not new energy

\begin{aligned}U_{\rm weak}&\simeq1+i\varphi\\H(0)&=i\\U_{\rm contrast}&\simeq i+i\varphi\\I_{\rm contrast}&\simeq1+2\varphi\end{aligned}

This illustrative weak-phase relation assumes a uniform background and a disk affecting only the zero order. The numerical model uses the actual finite illumination and finite disk, so halos and background leakage remain; it does not substitute this linear formula for the simulated image.

Passive matched correlation

\begin{aligned}H_{\rm match}&=P\frac{\widetilde G^*}{\max|\widetilde G|}\\U_{\rm corr}(x,y)&\propto\iint U(\xi,\eta)G^*(\xi+x,\eta+y)\,\mathrm d\xi\,\mathrm d\eta\end{aligned}

A smooth open-ring glyph is the programmed template. The same FFT data and passive normalized complex transmission calculate the correlation. Two repeated glyphs create peaks at inverted displacements. The disk gives a broad false response that can be brighter: an unnormalized positive-template correlation is not a classifier. This is an ideal complex mask, not a manufactured hologram or certified recognition system.

Sampling and physical scale

\begin{aligned}L_{\rm num}&=16\,\mathrm{mm},\quad N=512\\\Delta x&=L_{\rm num}/N\\\Delta\nu&=1/L_{\rm num}\\a_H&=\lambda f r_\nu\end{aligned}

Input illumination tapers to zero at the four-millimetre square boundary. Zero padding reduces circular wraparound but the discrete output remains periodic; hard masks are sensitive to frequency-grid placement. Samples are point intensity, not area-integrated camera pixels. Changing wavelength or focal length reprograms the mask to keep frequency coordinates fixed, so its physical dimensions change. Apparatus axial scale and transverse detail scales differ.

Common difficulties

An intensity photograph does not contain the entire optical field

Typical misconceptionFourier-transforming the input photograph reproduces the optical processing of any object.

Better mental modelThe lens acts on complex amplitude. A transparent phase object can have nearly uniform input intensity while a phase disk reveals its structure. This simulation knows the programmed complex transmission; it does not recover lost phase from a photograph.

Run the experiment

01
Scene 1: Blur versus edges
Compare open carrier, low pass and high pass on the target. The image inverts; high pass loses most power while emphasizing edges.
What to observe: Low pass softens fine detail; removing the central spectrum makes edges brighter relative to the background, not an energy gain.
02
Scene 2: Erase the stripe pair
In the stripe preset, click the paired spectrum peaks. Translate or widen the twin notch; stripe removal also discards overlapping object information.
What to observe: A horizontal Fourier slit passes horizontal frequency variations, hence vertical object stripes. Rotating the slit rotates the selected frequency direction.
03
Scene 3: Invisible phase becomes contrast
Compare phase cells with an open carrier and a quarter-wave disk. Reverse disk phase, then try matched correlation on the two glyphs.
What to observe: Phase filtering preserves power within the carrier; local brightness redistributes. Correlation peaks mark inverted displacements, not recovered object pixels.